#411799

order	512
nilpotency class	3
isoclinism	isoclinic class of #394283
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6)(7,16)(8,20)(9,25)(10,15)(11,14)(12,18)(13,23)(17,27)(19,29)(21,30)(22,31)(24,32)(26,28)(33,36)(34,35);(5,7,9,18,11,19,10,26)(6,12,15,16,14,28,25,29)(8,21,22,17,24,33,23,34)(13,30,32,35,31,36,20,27);(5,8)(6,13)(7,17)(9,22)(10,23)(11,24)(12,27)(14,31)(15,32)(16,30)(18,33)(19,34)(20,25)(21,26)(28,35)(29,36);(1,2,3,4)(6,14)(12,28)(13,31)(15,25)(16,29)(20,32)(27,35)(30,36)

Polycyclic presentation:

2^9|b^2=e,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318995);